May 6 2014
A Gaussian beam is a beam of electromagnetic radiation with intensity distributions, and a transverse electric field represented by Gaussian functions. Gaussian beams are named after the physicist and mathematician Johann Carl Friedrich Gauβ.
Following refraction by a diffraction-limited lens, a Gaussian beam is converted into another beam with a different set of parameters. The mathematical function representing the Gaussian beam is a resultant of the Helmholtz equation in paraxial form. This function is used to calculate the complex amplitude of the electric field of the beam.
Mathematical Expression
Gaussian beams are considered in conditions of small beam divergence, in order to apply paraxial approximation. A mathematical expression for the calculation of complex electric field amplitude of a monochromatic beam, propagating along the z-direction with wavelength λ, is given by:
E(r,z) = E0[w0/w(z)] exp (-r2/w(z)2) exp (-i [kz – arctan z/zR + kr2/2R(z)])
where:
E0 = peak amplitude
w0 = beam radius
k = wave number = 2π/λ
zR = Rayleigh number
R(z) = radius of curvature of wavefronts.
Parameters of Gaussian Beam
The geometry and nature of Gaussian beams are determined using a set of parameters, which includes the following:
- Beam divergence
- Radius of curvature
- Spot size, or beam width
- Rayleigh range and confocal parameter
- Gouy phase
- Complex beam parameter
Resonator Modes
Gaussian modes are the lowest order modes of an optical resonator, set along the transverse direction. A stable resonator will have homogeneous optical media, and a flat or parabolic surface between the media. Hence, lasers in the fundamental transverse mode often emit beams close to Gaussian shape.
Modes of high transverse order, on the other hand, are known as Laguerre-Gaussian, or Hermite-Gaussian, functions. Any deviation from a Gaussian beam shape can be estimated by the M2 factor. A Gaussian beam, having the highest beam quality, corresponds to M2 = 1.